Solve this equation. Can anybody do it?

Question

$$x / y = a + b \log(y)$$

Above is a relation between x and y. I have x. How do I find y ? Given: a and b are constants.

a is 0.710 while b is -0.014

Answer 1

For a given value of $x$, you want to solve for $y$ $$x / y = a + b ~ \log(y)$$ $a$ and $b$ being given constants. So, for example, let us write the equation as $$f(y)=y \Big( a + b ~ \log(y) \Big)-x$$ and you search $y$ such that $f(y)=0$.

A simple method for this kind of problem is Newton scheme which, starting from a guess $y_0$ will generate iterates according to $$y_{n+1}=y_n-\frac {f(y_n)}{f'(y_n)}$$ So, let us be very lazy and start iterating at $y_0=1$ using your values for $a$ and $b$ and $x=10$. The succesive iterates will then be $14.3477$, $14.8763$, $14.8765$ which is the solution for six significant figures.

Answer 2

The solution requires the Lambert W function.

$$ y = \exp\left(W(x e^{a/b}/b) - a/b\right)$$